"There is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds." G.H. Hardy, A Mathematician's Apology (1941)

It turns out that your friends having more friends than you do is a consequence of the fundamental structure of networks. It comes down to the difference between the average number of friends that any node in a network has versus the average number of friends of friends. Think of a party where the host invites 20 people, none of who know each other. The network that represents that party would look like one node in the center with 20 lines connecting it to otherwise unconnected nodes all around it. Now if you were to look at the average number of friends for any node in this network you have 20 nodes that have exactly 1 friend and one node that has exactly 20 friends, so you end up with an average of 40/21. So the average number of friends is just under two friends per node. Now the number of friends of friends is counted a bit differently. First you count the total number of friends of friends. In this cases all 21 nodes have a friends of friends number of 20. and therefore there are a total of 420 friends of friends in the network. Then you have to take that number of friends of friends and divide it by the total number of friendships in the network, which was reckoned above to be 40. This gives us an average number of friends of friends of 10.5, which is rather larger than the average number of friends. This is simply down to the host being counted 20 times in the friends of friends calculation, versus all of the guests only being counted once. It is similar in all networks, where the nodes with more connections are counted more often when figuring out the average number of friends of friends.

Of course being about averages there are always individuals in a network that really do have more friends than their friends do, but they are in the minority. So, it really does turn out that most of us have fewer friends than our friends do. Feld thought that this was important, and saw it as being related to a lot of other sociological phenomenon, specifically something that he and a collaborator, Bernard Grofman, called the “class size paradox”. The class size paradox is a known sociological phenomenon that observes that individual students at a college tend to experience an average class size that is larger than the actual average at the college. Feld had this conclusion about this research:

The tendency for most people to have fewer friends than their friends
have is one sociologically significant class size paradox. Individuals who
find themselves associated with people with more friends than they have
may conclude that they themselves are below average and somehow inad-
equate. The analysis presented in this paper indicates that most individu-
als have friends who have more friends than average and so provide an
unfair basis for comparison. Understanding the nature of a class size
paradox should help people to understand that their position is relatively
much better than their personal experiences have led them to believe.

This would have all probably stayed in the world of sociology, if Harvard’s Nicolas Christakis had not taken a look at Feld’s work and seen that it could have a very different application.

Nicolas Christakis is a physician and social scientist who, along with UCSD’s James Fowler, uses the tools of social network theory and sociology to help solve health problems. In a talk to TED in Cannes, France Christakis spoke about a new application of people’s friends having more friends than they do. He had been working on using social networks to help predict the spread and outbreak of epidemics, but he had been running into a problem. In Christakis’s own words, ‘The problem, however, is that mapping human social networks is not always possible. It can be expensive, not feasible, unethical, or, frankly, just not possible to do such a thing. So, how can we figure out who the central people are in a network without actually mapping the network?’ This is where Feld’s research comes in. Since Christakis was looking to help predict the spread and outbreak of epidemics he needed to be able to observe the most central and important nodes in a network. He knew that while random choice of people would not be the most useful tactic to use; perhaps if he took his random choices and asked them to nominate a friend they would be more central because, as he knew, they were likely to have more friends than his original choice. This idea would turn out to be rather potent.

Christakis used this method of friend nomination to study the spread of the swine flu, H1N1, through a population of Harvard students in 2009. He followed the people he had randomly chosen and the friends that they had nominated and within the friends group he received data that pointed towards an impending outbreak 46 days before the same indication of an outbreak appeared in the data from the randomly chosen group. That is a month and a half earlier warning of an epidemic, which could really help battle the spread of disease in the future. This is not just tied to epidemic research either, as a marketer could use this to help predict what trends will become hot before their competition has any idea or a pollster could see which presidential candidate is picking up steam before their campaign even has any idea. Christakis does say that the amount of notice that one would get from using this method would vary, but it turns out that Feld’s provocatively named paper is a more useful tool than Feld ever could have imagined.

2 Comments

Samuel, I know it’s your style and all, but if it wasn’t you I would have stopped at the end of the first paragraph. That last half sentence is the kind of statement I find very unhelpful for mathematics (to put it very, very diplomatically…).